To place the events in chronological order, follow this sequence:
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Immediately after Mexico gains independence in 1821, the Mexican government invites Americans to settle in its territory of Texas.
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American settlers in Texas rebel against Mexico and create an independent country called Texas.
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Mexico refuses to recognize the independence of Texas. Meanwhile, independent Texas asks the United States to annex it.
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Even though Mexico warns the United States that annexation would be considered an act of war, the United States annexes Texas anyway.